Wallis's Product/Proof 3
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Theorem
| \(\ds \prod_{n \mathop = 1}^\infty \frac {2 n} {2 n - 1} \cdot \frac {2 n} {2 n + 1}\) | \(=\) | \(\ds \frac 2 1 \cdot \frac 2 3 \cdot \frac 4 3 \cdot \frac 4 5 \cdot \frac 6 5 \cdot \frac 6 7 \cdot \frac 8 7 \cdot \frac 8 9 \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac \pi 2\) |
Proof
| \(\ds \prod_{n \mathop = 1}^\infty \frac {2 n} {2 n - 1} \cdot \frac {2 n} {2 n + 1}\) | \(=\) | \(\ds \prod_{n \mathop = 1}^\infty \frac {\paren {2 n - 0} \paren {2 n - 0} \times \dfrac 1 2 \times \dfrac 1 2} {\paren {2 n - 1} \paren {2 n + 1} \times \dfrac 1 2 \times \dfrac 1 2}\) | multiplying top and bottom by $\dfrac 1 2 \times \dfrac 1 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \prod_{n \mathop = 1}^\infty \frac {\paren {n - 0} \paren {n - 0} } {\paren {n - \dfrac 1 2} \paren {n + \dfrac 1 2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac { \map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 3 2} } {\map \Gamma {1} \map \Gamma {1} }\) | Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac { \sqrt \pi \times \dfrac {\sqrt \pi} 2 } {0! \times 0! }\) | Gamma Function of One Half, Gamma Difference Equation, Gamma Function Extends Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac \pi 2\) | Definition of Factorial |
$\blacksquare$
Source of Name
This entry was named for John Wallis.